Given a standard Brownian motion B µ = (B µ t ) 0≤t≤1 with drift µ ∈ IR , letting S µ t = max 0≤s≤t B µ s for t ∈ [0, 1] , and denoting by θ the time at which S µ 1 is attained, we consider the optimal prediction problem where the infimum is taken over all stopping times τ of B µ . Reducing the optimal prediction problem to a parabolic free-boundary problem and making use of local time-space calculus techniques, we show that the following stopping time is optimal: is a continuous decreasing

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... tion with b(1) = 0 that is characterised as the unique solution to a nonlinear Volterra integral equation. This also yields an explicit formula for V * in terms of b . If µ = 0 then there is a closed form expression for b . This problem was solved in [14] and [4] using the method of time change. The latter method cannot be extended to the case when µ = 0 and the present paper settles the remaining cases using a different approach. It is also shown that the shape of the optimal stopping set remains preserved for all Lévy processes. ( where the infimum is taken over all stopping times τ of B .